CN103840838A - Method for Bayes compressed sensing signal recovery based on self-adaptive measurement matrix - Google Patents

Abstract

The invention provides a method for Bayes compressed sensing signal recovery based on a self-adaptive measurement matrix and relates to the field of the information and communication technology. The method aims at solving the problem that an existing compressed sensing signal recovery method is low in accuracy. Based on the design of the self-adaptive measurement matrix in compressed sensing and combined with the Bayes compressed sensing algorithm, a design scheme of the compressed sensing method is obtained. The method is characterized in that the designed measurement matrix can be generated in a self-adaptive mode according to different signals, the purposes of determinacy and storage of the matrix are both achieved, and combined with the Bayes compressed sensing recovery algorithm of a relevant vector machine, the priority of a layered structure is introduced. The design scheme passes simulation verification, it is confirmed that the good signal recovery effect can be obtained, and the error range of recovered signals can be evaluated. The method is suitable for wireless signal transmission occasions in the information and communication technology.

Description

A kind of Bayes's compressed sensing signal recovery method based on adaptive observation matrix

Technical field

The present invention relates to Information & Communication Technology field, be specifically related to a kind of Bayes's compressed sensing signal recovery method.

Background technology

Compressed sensing technology can sample and can recover in high quality primary signal with very low sampling rate signal, solved the immense pressure of signal sampling, transmission and storage that people cause the great demand amount of information.The design of observing matrix and restoration methods is two parts very crucial in compressed sensing process.

Observing matrix is mainly divided into random observation matrix and certainty observing matrix.The recovery precision of random observation matrix is high, but its uncertain meeting bring difficulty to matrix stores and hardware realization; Certainty observing matrix can be saved memory space, be easy to hardware and realize, but its recovery effects is poor.Adaptive observation matrix is the nearly 2 years a kind of more novel observing matrix methods for designing that propose, and it is to generate corresponding observing matrix by the prior information of signal or sparse coefficient, and its performance will significantly be better than random observation matrix and certainty observing matrix.

The recovery algorithms of compressed sensing is mainly divided into protruding relaxed algorithm, greedy algorithm and combinational algorithm.Protruding relaxed algorithm just can obtain good reconstruct effect by a small amount of measured value, but amount of calculation is large; Greedy algorithm computation burden is little, but the measured value that successful reconstruct needs is many; The computational speed of combinational algorithm is fast, but needs a large amount of and difficult sampled value obtaining.Bayes method is to propose for 08 year, and it combines the advantage of above algorithm, and the recovery precision of signal is very high.

Does existing compressed sensing Technology Need solve: the noiseproof feature that how adds strong algorithms by observing matrix reasonable in design? does how the recovery algorithms of advanced design make in the situation that amount of calculation is very little with just restoring signal accurately of less measured value?

Summary of the invention

The present invention is the problem that the precision in order to solve existing compressed sensing signal recovery method is low, thereby a kind of Bayes's compressed sensing signal recovery method based on adaptive observation matrix is provided.

Bayes's compressed sensing signal recovery method based on adaptive observation matrix, it is realized by following steps:

Step 1, the observing matrix Φ ' that utilizes M × N to tie up, pass through formula:

y=Φ′f=Φ′Ψw=Φw （1）

Obtain the M × 1 dimension measured value y of N × 1 dimension unknown signaling f; M, N are positive integer, and M < < N; Φ is perception matrix; Ψ is sparse base;

Wherein: the unknown signaling f of N × 1 dimension is expressed as:

f=Ψw （2）

In formula: w is the sparse signal of N × 1 dimension;

For observing matrix Φ ' is designed to adaptive observation matrix, be specially:

According to formula (1), in time domain, due to the information that unknown signaling f has comprised primary signal, time domain observing matrix is Φ ';

In sparse territory, because sparse signal w has also comprised the information of primary signal, sparse territory observing matrix is Φ; Given this, it is feasible that the nonzero coefficient in sparse signal w is obtained to measured value, is specially:

Formula (1) is out of shape, obtains:

y=Φ′w=Φ′Ψ ^-1f=Φf （3）

At sparse base Ψ, be orthogonal in the situation that, formula (3) becomes:

y=Φ′w=Φ′Ψ ^Tf=Φf （4）

Now, the observing matrix of time domain becomes Φ, and the observing matrix in sparse territory becomes Φ ';

In sparse signal w, the number of nonzero value is M, and M is positive integer; In sparse signal w, the position of i nonzero value is j, 1≤i≤M; 1≤j≤N;

The middle element φ ' of observing matrix Φ ' _i,j=1, other element is all made as 0, as follows:

${\mathrm{\&phi;}}_{i,j}^{\&prime;}=\left\{\begin{array}{c}1,w_j\&NotEqual;0\\ 0,w_j=0\end{array}\right.---\left(5\right)$

Due to Φ=Φ ' Ψ ^t, the element therefore obtaining in Φ is:

φ _i,k=ψ _j,k （6）

In formula: 1≤k≤N; Here the Φ ' obtaining is the Φ in formula (1), and the Φ obtaining is the Φ ' in formula (1);

Step 2, the observation signal y that adopts the observing matrix Φ ' of step 1 acquisition by Bayes's compression sensing method, M × 1 to be tieed up carry out signal recovery, obtain the signal after recovering;

Be specially: because signal can produce noise in transmitting procedure, therefore the actual conditions of formula (2) should be:

y=Φ′f+n （7）

In formula, n is that average is 0, variances sigma ²unknown Gaussian noise;

According to sparse conversion coefficient, formula (7) is rewritten into following form:

y=Φ′Ψw+n=Φw+n （8）

Utilize the sparse property of w, the approximation of primary signal obtains by the optimization problem that solves following formula:

$\hat{w}={\arg }_w\min \{{\left|\right|y-\mathrm{\&Phi;w}\left|\right|}_2^2+\mathrm{\&tau;}{\left|\right|w\left|\right|}_0\}---\left(9\right)$

Wherein: || w|| ₀the l of sparse signal w ₀norm;

Use l ₁norm replaces l ₀norm, is converted into above formula:

$\hat{w}={\arg }_w\min \{{\left|\right|y-\mathrm{\&Phi;w}\left|\right|}_2^2+\mathrm{\&tau;}{\left|\right|w\left|\right|}_1\}---\left(10\right)$

Make w _srepresent M maximum value in a N dimensional vector w, a remaining N-M value is made as 0; Vector w _edivide and represent N-M element minimum in w, remaining element is set to 0;

Obtain thus:

w=w _s+w _e （11）

With

y=Φw=Φw _s+Φw _e=Φw _s+n _e （12）

In formula: n _e=Φ w _e;

According to Central Limit Theorem, n _ein element be 0 by an average Gaussian noise is approximate, consider the compressed sensing noise n that comprises in sampling process itself simultaneously _mtherefore, have:

y=Φw _s+n _e+n _m=Φw _s+n （13）

Gauss's likelihood model of measured value y is:

$p\left(y\right|w_s,{\mathrm{\&sigma;}}^2)={\left(2\mathrm{\&pi;}{\mathrm{\&sigma;}}^2\right)}^{-K/2}\exp (-\frac{1}{2{\mathrm{\&sigma;}}^2}{\left|\right|y-\mathrm{\&Phi;}{w}_s\left|\right|}^2)---\left(14\right)$

By estimating sparse vector w _swith noise variance σ ², the restoring signal of acquisition measured value y, the Bayes's compressed sensing signal completing based on adaptive observation matrix recovers.

In step 2, estimate sparse vector w _swith noise variance σ ²method be: adopt posterior probability density function method realize, be specially:

First, each the element priori in sparse signal w being defined as to average is 0 Gaussian Profile:

$p\left(w\right|\mathrm{\&alpha;})=\underset{i=1}{\overset{N}{\mathrm{\&Pi;}}}N\left(w_i\right|0,{\mathrm{\&alpha;}}_i^{-1})---\left(15\right)$

Wherein: α _iit is the precision of Gaussian probability-density function;

Then, make the priori of α obey Γ distribution:

$p\left(\mathrm{\&alpha;}\right|a,b)=\underset{i=1}{\overset{N}{\mathrm{\&Pi;}}}\mathrm{\&Gamma;}\left({\mathrm{\&alpha;}}_i\right|a,b)---\left(16\right)$

In conjunction with formula (15) and (16), obtain the priori probability density function of w:

$p\left(w\right|a,b)=\underset{i=1}{\overset{N}{\mathrm{\&Pi;}}}{\&Integral;}_0^{\&infin;}N\left(w_i\right|0,{\mathrm{\&alpha;}}_i^{-1})\mathrm{\&Gamma;}\left({\mathrm{\&alpha;}}_i\right|a,b)d{\mathrm{\&alpha;}}_i---\left(17\right)$

Wherein:

Γ (α _i| a, b) d α _iobey student-t(Student-t) distribute;

Suppose super parameter alpha and α ₀known, provide measured value y and matrix Φ, the posterior probability density function of w is analytically expressed as multivariable Gaussian Profile so, and its average and variance are:

μ=α ₀∑Φ ^Ty （18）

∑=(α ₀Φ ^TΦ+A) ^-1 （19）

Wherein: A=diag (α ₁, α ₂..., α _n);

At RVM(Relevance Vector Machine, Method Using Relevance Vector Machine) in, super parameter alpha and α ₀by Type-II type ML(Type-II Maximum Likelihood, II type maximum likelihood) process estimates, this approaching used α and α ₀point estimation ask the maximum of their marginal likelihood functions;

Application EM(Expectation-maximization, greatest hope) algorithm, obtain:

${\mathrm{\&alpha;}}_i^{\mathrm{new}}=\frac{{\mathrm{\&gamma;}}_i}{{\mathrm{\&mu;}}_i^2},i\&Element;\{\mathrm{1,2},...,N\}---\left(20\right)$

Wherein: μ _ii the average of calculating in (18) formula, wherein ∑ _iii the diagonal element that (19) formula calculates;

For noise variance σ ²=1/ α ₀, differential is estimated again:

$\frac{1}{{\mathrm{\&alpha;}}_0^{\mathrm{new}}}\stackrel{\mathrm{\&Delta;}}{=}\frac{{\left|\right|y-\mathrm{\&Phi;\&mu;}\left|\right|}_2^2}{K-\underset{i}{\mathrm{\&Sigma;}}{\mathrm{\&gamma;}}_i}---\left(21\right)$

Finally to w and α, α ₀the iterative computation that hockets, the result to the last obtaining convergence.

The present invention has adopted a kind of adaptive observing matrix in the design of observing matrix, and it is different from random observation matrix and certainty observing matrix.This adaptive observing matrix can not generate the in the situation that of signal the unknown, it is, according to the prior information of signal and sparse coefficient, the element in observing matrix is carried out to self adaptation adjustment, amplify signal component and suppressed noise component(s), so just greatly improved the noise robustness of algorithm and final signal and recovered precision.

Signal recovery algorithms of the present invention has adopted a kind of Bayes's compressed sensing algorithm based on Method Using Relevance Vector Machine, it has used Method Using Relevance Vector Machine to suppose priori estimated parameter on the basis of Bayesian frame, in consuming time and recovery effects, obtain good effect in the recovery of signal, and this algorithm can also provide the scope of error, this is that other compressed sensing recovery algorithms can not be accomplished.

Can obtain by emulation, use gaussian random matrix as observing matrix, in the time that signal to noise ratio is 20, the normalized mean squared error NMSE=0.045472 of BP algorithm; Signal to noise ratio is 5 o'clock, NMSE=0.39902.Use gaussian random matrix as observing matrix, in the time that signal to noise ratio is 20, the normalized mean squared error NMSE=0.054129 of OMP algorithm; Signal to noise ratio is 5 o'clock, NMSE=0.52724.Use the adaptive observation matrix of the present invention's design, in the time that signal to noise ratio is 20, the normalized mean squared error NMSE=0.002313 of the BCS algorithm based on RVM; Signal to noise ratio is 5 o'clock, NMSE=0.048724.The method has been controlled at recovery error in 5%, has had large increase than conventional method recovery error of 40% left and right in low signal-to-noise ratio situation.

The present invention has applied Bayes's compressed sensing signal recovery method on a kind of basis of adaptive observation matrix, finally obtained computing time short, recover precision high, noiseproof feature good and provide recovery error excellent results.

Accompanying drawing explanation

Fig. 1 is in the situation of SNR=20, primary signal analogous diagram, wherein signal length N=512, degree of rarefication K=20;

Fig. 2 is in the situation of SNR=20, noisy signal analogous diagram, wherein signal length N=512;

Fig. 3 is in the situation of SNR=20, uses random Gaussian matrix B P algorithm restoring signal analogous diagram, wherein signal length N=512, measured value M=100;

Fig. 4 is in the situation of SNR=5, noisy signal analogous diagram, wherein signal length N=512;

Fig. 5 is in the situation of SNR=5, uses random Gaussian matrix B P algorithm recovery effects analogous diagram, wherein signal length N=512, measured value M=100;

Fig. 6 is in the situation of SNR=20, uses adaptive matrix BP algorithm recovery effects analogous diagram of the present invention, wherein signal length N=512, measured value M=100;

Fig. 7 is in the situation of SNR=5, uses adaptive matrix BP algorithm recovery effects analogous diagram of the present invention, wherein signal length N=512, measured value M=100;

Fig. 8 is the level prior model of the Bayes's compressed sensing algorithm based on Method Using Relevance Vector Machine;

Fig. 9 is in the situation of SNR=20, uses random Gaussian matrix OMP algorithm recovery effects analogous diagram, wherein signal length N=512, measured value M=100;

Figure 10 is in the situation of SNR=5, uses random Gaussian matrix OMP algorithm recovery effects analogous diagram, wherein signal length N=512, measured value M=100;

Figure 11 is in the situation of SNR=20, uses the Bayes compressed sensing algorithm recovery effects analogous diagram of random Gaussian matrix based on Method Using Relevance Vector Machine, wherein signal length N=512, measured value M=100;

Figure 12 is in the situation of SNR=5, uses the Bayes compressed sensing algorithm recovery effects analogous diagram of random Gaussian matrix based on Method Using Relevance Vector Machine, wherein signal length N=512, measured value M=100;

Embodiment

Embodiment one, a kind of Bayes's compressed sensing signal recovery method based on adaptive observation matrix,

Compressive sensing theory comprises three following steps:

1), the unknown signaling f of N × 1 dimension is sparse under linear base Ψ (N × N), that is:

f=Ψw （2）

Wherein: w is the sparse signal of N × 1 dimension, and its most of coefficient is all 0;

2), utilize the observing matrix Φ ' of M × N dimension to obtain measured value:

y=Φ′f=Φ′Ψw=Φw （1）

Wherein: y is the measured value of M × 1 dimension, Φ=Φ ' Ψ is the perception matrix of M × N dimension;

3) known Φ ', Ψ, y, selects suitable recovery algorithms to recover f:

$\hat{f}={\mathrm{\&Phi;}}^{\&prime;-1}y$

1, the method for designing of observing matrix

The design of observing matrix Φ ' of the present invention has adopted a kind of adaptive method.According to formula (1), matrix Φ ' can be called to time domain observing matrix, matrix Φ is called to sparse territory observing matrix.

In time domain, the information that signal f has comprised primary signal; In sparse territory, vectorial w has also comprised the information of primary signal.Given this, getting nonzero coefficient in w, to obtain measured value be also feasible.Formula (1) is out of shape, obtains:

y=Φ′w=Φ′Ψ ^-1f=Φf （3）

If sparse base Ψ is orthogonal, formula (3) can become:

y=Φ′w=Φ′Ψ ^Tf=Φf （4）

Concerning the observation of time domain, observing matrix just becomes Φ like this.

Due to Φ=Φ ' Ψ ^t, sparse territory is determined, Ψ ^talso just determine, what therefore mainly need analysis is the design of Φ '.Observation type (4) discovery, measured value y is actually the nonzero value in sparse vector w, and therefore the object of matrix Φ ' is extracted nonzero value exactly from w.As long as find the position of nonzero value in w, just can determine the composition of matrix Φ '.

In vector w, the number of nonzero value is M, and the position of the individual nonzero value of i in w (1≤i≤M) is j (1≤j≤N), the middle element φ ' of matrix Φ ' _i,j=1, other element is all made as 0, as follows:

${\mathrm{\&phi;}}_{i,j}^{\&prime;}=\left\{\begin{array}{c}1,w_j\&NotEqual;0\\ 0,w_j=0\end{array}1\&le;i\&le;M,1\&le;j\&le;N\right.---\left(6\right)$

Element in matrix Φ ' only has 0 and 1, and wherein 1 number is M.After matrix Φ ' obtains, the element of matrix Φ also just can obtain.Due to Φ=Φ ' Ψ ^t, the element therefore obtaining in Φ is:

φ _i,k=ψ _j,k1≤i≤M,1≤j≤N,1≤k≤N （7）

Next if time-domain signal f is observed, use so Φ to obtain measured value y, if sparse coefficient vector w is observed, use so Φ ' to obtain measured value y.Because observing matrix is determined by signal, the observing matrix that different signals obtains is different, and the observing matrix of therefore constructing in the present invention is called adaptive observation matrix.

Fig. 3,5 and Fig. 6,7 be respectively use random Gaussian observing matrix and while using adaptive observation matrix of the present invention the signal of BP algorithm recover analogous diagram, we can see, the recovery error that uses adaptive observation matrix to obtain will be far smaller than the error that uses random Gaussian observing matrix to produce.

2, the method for designing of recovery algorithms

What the recovery algorithms in the present invention adopted is Bayes (BCS) the compressed sensing algorithm based on Method Using Relevance Vector Machine (RVM).Because signal can produce noise in transmitting procedure, therefore the ideal situation of formula (1) should change into

y=Φ′f+n （7）

We can be write formula (8) as following form according to sparse conversion coefficient:

y=Φ′Ψw+n=Φw+n （8）

According to compressive sensing theory, in the time that the quantity of measured value is less than the quantity of signal coefficient (M < < N), use suitable recovery algorithms just can determine and in situation, accurately recover initialize signal f at some.Due to M < < N, the inverse problem of direct solution formula (9) is an ill-conditioning problem, cannot direct solution.

Utilize the sparse property of w, the approximation of primary signal can obtain by the optimization problem that solves following formula:

$\hat{w}={\arg }_w\min \{{\left|\right|y-\mathrm{\&Phi;w}\left|\right|}_2^2+\mathrm{\&tau;}{\left|\right|w\left|\right|}_0\}\&CenterDot;---\left(10\right)$

Wherein || w|| ₀the l of w ₀norm.This optimization problem is a np hard problem, therefore needs to simplify, and the most frequently used method is to use l ₁norm replaces l ₀norm, this optimization problem is just converted into following formula:

$\hat{w}={\arg }_w\min \{{\left|\right|y-\mathrm{\&Phi;w}\left|\right|}_2^2+\mathrm{\&tau;}{\left|\right|w\left|\right|}_0\}\&CenterDot;---\left(11\right)$

We allow w _srepresent M maximum value in a N dimensional vector w, a remaining N-M value is made as 0.Similarly, vectorial w _edivide and represent N-M element minimum in w, remaining element is set to 0.Obtain thus:

w=w _s+w _e （11）

With:

y=Φw=Φw _s+Φw _e=Φw _s+n _e （12）

Wherein n _e=Φ w _e;

Because Φ obtains by stochastical sampling, so according to Central Limit Theorem, n _ein element can be 0 by an average Gaussian noise is approximate, consider the compressed sensing noise n that comprises in sampling process itself simultaneously _mso, have:

y=Φw _s+n _e+n _m=Φw _s+n （13）

In formula, n is that average is 0, variances sigma ²unknown Gaussian noise.Gauss's likelihood model of y is:

$p\left(y\right|w_s,{\mathrm{\&sigma;}}^2)={\left(2\mathrm{\&pi;}{\mathrm{\&sigma;}}^2\right)}^{-K/2}\exp (-\frac{1}{2{\mathrm{\&sigma;}}^2}{\left|\right|y-\mathrm{\&Phi;}{w}_s\left|\right|}^2)---\left(14\right)$

Through analysis above, compressed sensing problem changes into w _sthe linear regression problem of sparse constraint.

Suppose known Φ, that need estimation is sparse vector w _swith noise variance σ ², need to find their posterior probability density function.

In RVM, introduce the priori of layering, it has similar character to Laplace priori, but it allows conjugate exponent analysis.First, each the element priori in w being defined as to average is 0 Gaussian Profile:

$p\left(w\right|\mathrm{\&alpha;})=\underset{i=1}{\overset{N}{\mathrm{\&Pi;}}}N\left(w_i\right|0,{\mathrm{\&alpha;}}_i^{-1})---\left(15\right)$

Wherein: α _iit is the precision of Gaussian probability-density function.Then the priori of, supposing α is obeyed Γ distribution:

$p\left(\mathrm{\&alpha;}\right|a,b)=\underset{i=1}{\overset{N}{\mathrm{\&Pi;}}}\mathrm{\&Gamma;}\left({\mathrm{\&alpha;}}_i\right|a,b)---\left(16\right)$

In conjunction with formula (15) and (16), obtain the priori of w

$p\left(w\right|a,b)=\underset{i=1}{\overset{N}{\mathrm{\&Pi;}}}{\&Integral;}_0^{\&infin;}N\left(w_i\right|0,{\mathrm{\&alpha;}}_i^{-1})\mathrm{\&Gamma;}\left({\mathrm{\&alpha;}}_i\right|a,b)d{\mathrm{\&alpha;}}_i---\left(17\right)$

Wherein:

Γ (α _i| a, b) d α _iobeying Student-t distributes.

In the time that the selection of a and b is suitable, Student-t is distributed in w _i=0 place obtains peak value, and therefore the priori in (17) formula has promoted the sparse property of w.Similarly, can select Γ priori for noise variance.

Suppose super parameter alpha and α ₀known, provide measured value y and matrix Φ, the posteriority of w can analytically be expressed as multivariable Gaussian Profile so, and its average and variance are:

μ=α ₀∑Φ ^Ty （18）

∑=(α ₀Φ ^TΦ+A) ^-1 （19）

Wherein: A=diag (α ₁, α ₂..., α _n).

In RVM, super parameter alpha and α ₀estimate by Type-II type ML process, this approaching used α and α ₀point estimation ask the maximum of their marginal likelihood functions.Application EM algorithm, obtains:

${\mathrm{\&alpha;}}_i^{\mathrm{new}}=\frac{{\mathrm{\&gamma;}}_i}{{\mathrm{\&mu;}}_i^2},i\&Element;\{\mathrm{1,2},...,N\}---\left(20\right)$

Wherein: μ _ii the average of calculating in (18) formula,

wherein ∑ _iii the diagonal element that (19) formula calculates.

For noise variance σ ²=1/ α ₀, differential is estimated again:

$\frac{1}{{\mathrm{\&alpha;}}_0^{\mathrm{new}}}\stackrel{\mathrm{\&Delta;}}{=}\frac{{\left|\right|y-\mathrm{\&Phi;\&mu;}\left|\right|}_2^2}{K-\underset{i}{\mathrm{\&Sigma;}}{\mathrm{\&gamma;}}_i}---\left(21\right)$

Then to w and α, α ₀the iterative computation that hockets, the result to the last obtaining convergence.

The level prior model of this algorithm as shown in Figure 8.Fig. 9,10,11 and 12 is respectively that the signal of OMP algorithm and the BCS algorithm based on RVM recovers analogous diagram in the situation that of same use random Gaussian observing matrix.Observation can obtain, and the BCS algorithm based on RVM will be higher than OMP algorithm in the recovery precision of signal, and the estimation of error range is provided.

The present invention has following characteristics and marked improvement:

1, the present invention, with the basis that is designed to of adaptive observation matrix in compressed sensing, obtains a kind of design of compression sensing method in conjunction with Bayes's compressed sensing algorithm.The observing matrix of its design can generate adaptively according to unlike signal, has obtained extraordinary signal recovery effects in conjunction with Bayes's compressed sensing recovery algorithms, and can estimate the error range of restoring signal.

2, in compressed sensing scheme of the present invention, the design of observing matrix is to generate corresponding observing matrix by the prior information of sparse coefficient, and its performance will significantly be better than other random observation matrixes and certainty observing matrix.

3, the present invention has introduced the theory of Bayesian inference and Method Using Relevance Vector Machine in the design of compressed sensing recovery algorithms, makes algorithm combine the advantage of protruding relaxed algorithm and greedy algorithm, and signal recovers precision and computing is consuming time has all reached satisfied result.

4, in the design of adaptive observation matrix of the present invention, need to find out the nonzero value in sparse vector, then the element of relevant position in observing matrix is set to 1, other element is made as 0, and this noiseproof feature for algorithm is very helpful.Find by emulation, compared with other observing matrix, this adaptive observing matrix recovers precision significant raising.

In 5 Bayes's recovery algorithms of the present invention, introduce the structure of layering priori, and super parameter has wherein been estimated with Method Using Relevance Vector Machine.Find by emulation, compared with other recovery algorithms, this recovery algorithms that the present invention proposes is higher to the recovery precision of signal, and the error range of signal can be provided.

Claims (2)

1. the Bayes's compressed sensing signal recovery method based on adaptive observation matrix, is characterized in that: it is realized by following steps:

Step 1, the observing matrix Φ ' that utilizes M × N to tie up, pass through formula:

y=Φ′f=Φ′Ψw=Φw （1）

Obtain the M × 1 dimension measured value y of N × 1 dimension unknown signaling f; M, N are positive integer, and M < < N; Φ is perception matrix; Ψ is sparse base;

Wherein: the unknown signaling f of N × 1 dimension is expressed as:

f=Ψw （2）

In formula: w is the sparse signal of N × 1 dimension;

For observing matrix Φ ' is designed to adaptive observation matrix, be specially:

According to formula (1), in time domain, due to the information that unknown signaling f has comprised primary signal, time domain observing matrix is Φ ';

In sparse territory, because sparse signal w has also comprised the information of primary signal, sparse territory observing matrix is Φ;

Nonzero coefficient in sparse signal w is obtained to measured value, is specially:

Formula (1) is out of shape, obtains:

y=Φ′w=Φ′Ψ ^-1f=Φf （3）

At sparse base Ψ, be orthogonal in the situation that, formula (3) becomes:

y=Φ′w=Φ′Ψ ^Tf=Φf （4）

Now, the observing matrix of time domain becomes Φ, and the observing matrix in sparse territory becomes Φ ';

In sparse signal w, the number of nonzero value is M, and M is positive integer; In sparse signal w, the position of i nonzero value is j, 1≤i≤M; 1≤j≤N;

The middle element φ ' of observing matrix Φ ' _i,j=1, other element is all made as 0, as follows:

${\mathrm{\&phi;}}_{i,j}^{\&prime;}=\left\{\begin{array}{c}1,w_j\&NotEqual;0\\ 0,w_j=0\end{array}\right.---\left(5\right)$

Due to Φ=Φ ' Ψ ^t, the element therefore obtaining in Φ is:

φ _i,k=ψ _j,k （6）

In formula: 1≤k≤N; Here the Φ ' obtaining is the Φ in formula (1), and the Φ obtaining is the Φ ' in formula (1);

Step 2, the observation signal y that adopts the observing matrix Φ ' of step 1 acquisition by Bayes's compression sensing method, M × 1 to be tieed up carry out signal recovery, obtain the signal after recovering;

Be specially: because signal can produce noise in transmitting procedure, therefore the actual conditions of formula (2) should be:

y=Φ′f+n （7）

In formula, n is that average is 0, variances sigma ²unknown Gaussian noise;

According to sparse conversion coefficient, formula (7) is rewritten into following form:

y=Φ′Ψw+n=Φw+n （8）

Utilize the sparse property of w, the approximation of primary signal obtains by the optimization problem that solves following formula:

$\hat{w}={\arg }_w\min \{{\left|\right|y-\mathrm{\&Phi;w}\left|\right|}_2^2+\mathrm{\&tau;}{\left|\right|w\left|\right|}_0\}---\left(9\right)$

Wherein: || w|| ₀the l of sparse signal w ₀norm;

Use l ₁norm replaces l ₀norm, is converted into above formula:

$\hat{w}={\arg }_w\min \{{\left|\right|y-\mathrm{\&Phi;w}\left|\right|}_2^2+\mathrm{\&tau;}{\left|\right|w\left|\right|}_1\}---\left(10\right)$

Make w _srepresent M maximum value in a N dimensional vector w, a remaining N-M value is made as 0; Vector w _erepresent N-M element minimum in w, remaining element is set to 0;

Obtain thus:

w=w _s+w _e （11）

With

y=Φw=Φw _s+Φw _e=Φw _s+n _e （12）

In formula: n _e=Φ w _e;

According to Central Limit Theorem, n _ein element be 0 by an average Gaussian noise is approximate, consider the compressed sensing noise n that comprises in sampling process itself simultaneously _mtherefore, have:

y=Φw _s+n _e+n _m=Φw _s+n （13）

Gauss's likelihood model of measured value y is:

$p\left(y\right|w_s,{\mathrm{\&sigma;}}^2)={\left(2\mathrm{\&pi;}{\mathrm{\&sigma;}}^2\right)}^{-K/2}\exp (-\frac{1}{2{\mathrm{\&sigma;}}^2}{\left|\right|y-\mathrm{\&Phi;}{w}_s\left|\right|}^2)---\left(14\right)$

By estimating sparse vector w _swith noise variance σ ², the restoring signal of acquisition measured value y, the Bayes's compressed sensing signal completing based on adaptive observation matrix recovers.

2. a kind of Bayes's compressed sensing signal recovery method based on adaptive observation matrix described in 1 as requested, is characterized in that estimating in step 2 sparse vector w _swith noise variance σ ²method be: adopt posterior probability density function method realize, be specially:

First, each the element priori in sparse signal w being defined as to average is 0 Gaussian Profile:

$p\left(w\right|\mathrm{\&alpha;})=\underset{i=1}{\overset{N}{\mathrm{\&Pi;}}}N\left(w_i\right|0,{\mathrm{\&alpha;}}_i^{-1})---\left(15\right)$

Wherein: α _iit is the precision of Gaussian probability-density function;

Then, make the priori of α obey Γ distribution:

$p\left(\mathrm{\&alpha;}\right|a,b)=\underset{i=1}{\overset{N}{\mathrm{\&Pi;}}}\mathrm{\&Gamma;}\left({\mathrm{\&alpha;}}_i\right|a,b)---\left(16\right)$

In conjunction with formula (15) and (16), obtain the priori probability density function of w:

$p\left(w\right|a,b)=\underset{i=1}{\overset{N}{\mathrm{\&Pi;}}}{\&Integral;}_0^{\&infin;}N\left(w_i\right|0,{\mathrm{\&alpha;}}_i^{-1})\mathrm{\&Gamma;}\left({\mathrm{\&alpha;}}_i\right|a,b)d{\mathrm{\&alpha;}}_i---\left(17\right)$

Wherein:

Γ (α _i| a, b) d α _iobeying Student-t distributes;

Suppose super parameter alpha and α ₀known, provide measured value y and matrix Φ, the posterior probability density function of w is analytically expressed as multivariable Gaussian Profile so, and its average and variance are:

μ=α ₀∑Φ ^Ty （18）

∑=(α ₀Φ ^TΦ+A) ^-1 （19）

Wherein: A=diag (α ₁, α ₂..., α _n);

In Method Using Relevance Vector Machine RVM, super parameter alpha and α ₀estimate by Type-II type maximum likelihood process, this approaching used α and α ₀point estimation ask the maximum of their marginal likelihood functions;

Application greatest hope algorithm, obtains:

${\mathrm{\&alpha;}}_i^{\mathrm{new}}=\frac{{\mathrm{\&gamma;}}_i}{{\mathrm{\&mu;}}_i^2},i\&Element;\{\mathrm{1,2},...,N\}---\left(20\right)$

Wherein: μ _ii the average of calculating in (18) formula,

wherein ∑ _iii the diagonal element that (19) formula calculates;

For noise variance σ ²=1/ α ₀, differential is estimated again:

$\frac{1}{{\mathrm{\&alpha;}}_0^{\mathrm{new}}}\stackrel{\mathrm{\&Delta;}}{=}\frac{{\left|\right|y-\mathrm{\&Phi;\&mu;}\left|\right|}_2^2}{K-\underset{i}{\mathrm{\&Sigma;}}{\mathrm{\&gamma;}}_i}---\left(21\right)$

Finally to w and α, α ₀the iterative computation that hockets, the result to the last obtaining convergence.

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